Abstract
We examine the impact ofevenness (all cuts having even free capacity) andlocal evenness (cuts that separate a single vertex having even free capacity) on homotopic knock-knee routing. Kaufmann and Mehlhorn have presented a linear-time algorithm for routing even instances. We show that routing locally even instances is NP-hard. If we are permitted to move modules slightly, however, then we can efficiently route any locally even instance in which the free capacity of every cut is nonnegative. This fact implies that locally even instances can be one-dimensionally compacted in polynomial time. But when the assumption of local evenness is dropped, routing again becomes NP-hard, whether or not modules may move.
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References
M. Becker and K. Mehlhorn, Algorithms for routing in planar graphs,Ada Informatica, Vol. 23, No. 2 (1986), pp. 163–176.
M. L. Brady and D. J. Brown, Arbitrary planar routing with four layers,1984 Conference on Advanced Research in VLSI, MIT (January 1984), pp. 194–201.
R. Cole and A. Siegel, River routing every which way, but loose,25th Annual Symposium on Foundations of Computer Science (October 1984), pp. 65–73.
A. Frank, Disjoint paths in a rectilinear grid,Combinatorica, Vol. 2, No. 4 (1982), pp. 361–371.
S. Gao, M. Jerrum, M. Kaufmann, K. Mehlhorn, W. Rülling, and C. Storb, On continuous homotopic one layer routing,Fourth Annual Symposium on Computational Geometry (June 1988), pp. 392–402.
S. Gao, M. Kaufmann, and F. M. Maley, Advances in homotopic layout compaction,1989 ACM Symposium on Parallel Algorithms and Architectures (June 1989), pp. 273–282.
M. Kaufmann and K. Mehlhorn, On local routing of two-terminal nets, Technical Report, SFB 124, Universität des Saarlandes (1986); submitted toJournal of Combinatorial Theory Series B.
M. Kaufmann and K. Mehlhorn, Routing through a generalized switchbox,Journal of Algorithms, Vol. 7, No. 4 (1986), pp. 510–531.
M. Kaufmann and K. Mehlhorn, A linear-time algorithm for the local routing problem, submitted toSIAM Journal on Computing.
M. R. Kramer and J. van Leeuwen, Wire-routing is NP-complete, Technical Report RUU-CS-82-4, Department of Computer Science, University of Utrecht, the Netherlands (February 1982).
C. E. Leiserson and F. M. Maley, Algorithms for routing and testing routability of planar VLSI layouts,17th Annual ACM Symposium on Theory of Computing (May 1985), pp. 69–78.
W. Lipski, Jr., On the structure of three-layer wirable layouts, inAdvances in Computing Research, Vol. 2 (edited by F. P. Preparata), JAI Press, Greenwich, CT, 1984, pp. 231–243.
F. M. Maley, Compaction with automatic jog introduction,1985 Chapel Hill Conference on VLSI (May 1985), pp. 261–283.
F. M. Maley, Toward a mathematical theory of single-layer wire routing,Fifth MIT Conference on Advanced Research in VLSI (March 1988), pp. 277–296.
F. M. Maley,Single-Layer Wire Routing and Compaction, MIT Press, Cambridge, MA (1989).
F. M. Maley, A generic algorithm for one-dimensional homotopic compaction,Algorithmica, to appear.
K. Matsumoto, T. Nishizeki, and N. Saito, An efficient algorithm for finding multi-commodity flows in planar networks,SIAM Journal on Computing, Vol. 15, No. 2 (1986), pp. 495–510.
K. Mehlhorn and F. P. Preparata, Routing through a rectangle,Journal of the ACM, Vol. 33, No. 1 (1986), pp. 60–85.
T. Nishizeki, N. Saito, and K. Suzuki, A linear-time routing algorithm for convex grids,IEEE Transactions on Computer-Aided Design, Vol. CAD-4, No. 1 (1985), pp. 68–76.
H. Okamura and P. Seymour, Multicommodity flows in planar graphs,Journal of Combinatorial Theory Series B, Vol. 31 (1981), pp. 75–81.
F. P. Preparata and W. Lipski, Jr., Optimal three-layer channel routing,IEEE Transactions on Computers, Vol. C-33 (1984), pp. 350–357.
A. Schrijver, Decomposition of graphs on surfaces and a homotopic circulation theorem, preprint.
A. Schrijver, Edge-disjoint homotopic paths in straight-line planar graphs, preprint.
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Communicated by Kurt Mehlhorn.
This work was supported in part by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 124, Teilprojekt B2 (VLSI Entwurf und Parallelität), and in part by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, Grant NSF-STC88-09648. Miller Maley was also supported by a Mathematical Sciences Postdoctoral Research Fellowship from the National Science Foundation, Grant DMS-8705835.
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Kaufmann, M., Maley, F.M. Parity conditions in homotopic knock-knee routing. Algorithmica 9, 47–63 (1993). https://doi.org/10.1007/BF01185338
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DOI: https://doi.org/10.1007/BF01185338